This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
The problem states that p varies directly as q and inversely as the square of r. This can be written as a proportionality: p (q)/(r^2) To convert this into an equation, we introduce a constant of proportionality, k: p = k (q)/(r^2) We are given a table of values for p, q, and r. a) Find the value of K Step 1: Use the first set of values from the table to find the constant k. Given p=9, q=6, r=3. Substitute these values into the equation: 9 = k (6)/(3^2) 9 = k (6)/(9) 9 = k (2)/(3) Step 2: Solve for k. Multiply both sides by (3)/(2): k = 9 × (3)/(2) k = (27)/(2) k = 13.5 The value of K is (27)/(2) or 13.5. b) Find the value of a Step 1: Use the second set of values from the table: p=12, q=72, r=a. Substitute these values and the calculated k = (27)/(2) into the equation p = k (q)/(r^2): 12 = (27)/(2) (72)/(a^2) Step 2: Simplify the right side of the equation. 12 = (27 × 72)/(2a^2) 12 = (1944)/(2a^2) 12 = (972)/(a^2) Step 3: Solve for a^2. Multiply both sides by a^2: 12a^2 = 972 Divide by 12: a^2 = (972)/(12) a^2 = 81 Step 4: Solve for a. Take the square root of both sides: a = ±sqrt(81) a = ± 9 The value of a is ± 9. c) Find the positive value of b Step 1: Use the third set of values from the table: p=b, q=42, r=3. Substitute these values and the calculated k = (27)/(2) into the equation p = k (q)/(r^2): b = (27)/(2) (42)/(3^2) Step 2: Simplify the expression for b. b = (27)/(2) (42)/(9) b = (27)/(2) × (14)/(3) b = (27 × 14)/(2 × 3) b = (378)/(6) b = 63 The positive value of b is 63.